Ladder contours are undetectable in the periphery:A crowding effect?

McGill Vision Research Unit, Department of Ophthalmology,McGill University, Montréal, Québec, CanadaKeith A. May

McGill Vision Research Unit, Department of Ophthalmology,McGill University, Montréal, Québec, CanadaRobert F. Hess

We studied the perceptual integration of contours consisting of Gabor elements positioned along a smooth path,embedded among distractor elements. Contour elements either formed tangents to the path (“snakes”) or wereperpendicular to it (“ladders”). Perfectly straight snakes and ladders were easily detected in the fovea but, at aneccentricity of 6-, only the snakes were detectable. The disproportionate impairment of peripheral ladder detectionremained when we brought foveal performance away from ceiling by jittering the orientations of the elements. Wepropose that the failure to detect peripheral ladders is a form of crowding, the phenomenon observed when identi!cationof peripherally located letters is disrupted by "anking letters. D. G. Pelli, M. Palomares, and N. J. Majaj (2004) outlined amodel in which simple feature detectors are followed by integration !elds, which are involved in tasks, such as letteridenti!cation, that require the outputs of several detectors. They proposed that crowding occurs because small integration!elds are absent from the periphery, leading to inappropriate feature integration by large peripheral integration !elds. Weargue that the “association !eld,” which has been proposed to mediate contour integration (D. J. Field, A. Hayes, & R. F.Hess, 1993), is a type of integration !eld. Our data are explained by an elaboration of Pelli et al.’s model, in whichweak ladder integration competes with strong snake integration. In the fovea, the association !elds were small, andthe model integrated snakes and ladders with little interference. In the periphery, the association !elds were large,and integration of ladders was severely disrupted by interference from spurious snake contours. In contrast, themodel easily detected snake contours in the periphery. In a further demonstration of the possible link betweencontour integration and crowding, we ran our contour integration model on groups of three-letter stimuli made fromshort line segments. Our model showed several key properties of crowding: The critical spacing for crowding to occurwas independent of the size of the target letter, scaled with eccentricity, and was greater on the peripheral side ofthe target.

Keywords: contour integration, fovea, periphery, crowding, snakes, laddersCitation: May, K. A., & Hess, R. F. (2007). Ladder contours are undetectable in the periphery: A crowding effect? Journal ofVision, 7(13):9, 1–15, http://journalofvision.org/7/13/9/, doi:10.1167/7.13.9.

Introduction

The means by which low-level features in the visualenvironment are grouped to create high-level structure hasbeen studied for more than a century (e.g., Ehrenfels,1890/1988; Wertheimer, 1923/1938) but remains largelymysterious. In this paper, we attempt to shed some lighton the grouping processes that underlie the perceptualintegration of contours.Pelli, Palomares, and Majaj (2004) outlined a two-stage

model of feature integration, in which the outputs ofsimple feature detectors are combined using integrationfields. They proposed that the integration fields areinvolved in any task which depends on the outputs ofmore than one simple feature detector. One such task isletter identification. Pelli et al. argued that the phenom-enon of crowding (whereby identification of a peripherallylocated target letter is impaired by the presence of flanking

letters) is caused by inappropriate integration of featuresfrom target and flankers due to the absence of smallintegration fields in the periphery.Pelli et al. (2004) argued that, at any point in the visual

field, the minimum integration field size is proportional toeccentricity. The evidence for this is that the criticaltarget-mask spacing for crowding to occur is about halfthe eccentricity for several quite different tasks: letteridentification (Bouma, 1970; Pelli et al., 2004; Toet &Levi, 1992), fine discrimination of line orientation(Andriessen & Bouma, 1976), and fine discrimination ofcontrast, orientation, and spatial frequency of gratings(Wilkinson, Wilson, & Ellemberg, 1997).Contour detection is another task that requires the

outputs of several simple feature detectors to be integrated(Field, Hayes, & Hess, 1993). Field et al. (1993) proposedthat the elements along a smooth contour are integrated byan association field. We consider that the association fieldis an example of Pelli et al.’s (2004) integration field.

Journal of Vision (2007) 7(13):9, 1–15 http://journalofvision.org/7/13/9/ 1

doi: 10 .1167 /7 .13 .9 Received December 12, 2006; published October 29, 2007 ISSN 1534-7362 * ARVO

In most studies of contour integration, the contourelements have formed tangents to the path of the contour.Performance is very poor when the elements are orientedat 45- to the path (Ledgeway, Hess, & Geisler, 2005) butcan be reasonably high when the elements are orthogonalto it (Bex, Simmers, & Dakin, 2001; Field et al., 1993;Hess, Ledgeway, & Dakin, 2000; Ledgeway et al., 2005).The contour types with elements tangential and orthogo-nal to the path were termed “snakes” and “ladders,”respectively, by Bex et al. (2001). Having found thatladders were harder to detect than snakes, Field et al.(1993) concluded that “the association between elementsis stronger along the axis of the element than orthogonalto the axis” (p. 185).We therefore have reasonable grounds for inferring two

characteristics of the association fields: (1) the minimumfield size at any point in the visual field scales witheccentricity; and (2) the associations are stronger forsnake than ladder configurations. In this paper, weconsider how these two characteristics might combine todetermine the detectability of snake and ladder contours inthe periphery.Let us consider two broad classes of association field

model. In one class of model, snakes and ladders areintegrated by different types of association field, so thatintegration proceeds independently for the two types ofcontour. If the minimum field size scales with eccen-tricity in the same way for both types of association field(as appears to be the case for other types of integrationfield), then we might expect that peripheral viewingwould have a roughly similar effect on each type ofcontour. In the other class of model, snakes and laddersare integrated by the same association field, and “snake”and “ladder” associations compete directly against eachother for ownership of the elements. In this second classof model, if ladder associations were weaker than snakeassociations (as concluded by Field et al., 1993), then wewould expect peripheral viewing to cause a much greaterimpairment of ladder detection than snake detection. Thisis because, as the association field size gets larger, there isan increased probability that a ladder element will beapproximately collinear with a distractor within the samefield; if snake associations are stronger than ladderassociations, then the ladder element is likely to form asnake with the collinear distractor element, preventing itfrom forming part of the ladder contour. Straight snakecontours, on the other hand, would be relatively undis-turbed by increases in association field size because, evenwith large association fields, the snake elements would bemost strongly linked to their neighbors along the contour.Later, we present modeling results that confirm theintuitive argument presented here.Our argument that snake detection would be relatively

unimpaired in the periphery only applies to snakes thatare fairly straight. We would expect detection of snakeswith substantial curvature to be impaired in the periph-ery, for the same reason as ladders: If the snake elements

are not completely collinear then, as the field sizeincreases, there is an increased probability that anelement of a curved snake will be more collinear witha distractor element within the field than with itsneighbors along the contour. The snake element maythen form an association with the distractor element,interrupting the integration of the target contour. Thesepredictions relating to straight and curved peripheralsnakes have already been confirmed: In the periphery,performance on curved snakes tends to drop moresubstantially than on straight snakes (Hess & Dakin,1997, 1999; Nugent, Keswani, Woods, & Peli, 2003). Thisfinding would be predicted by both classes of associationfield model outlined above. The two classes of modeldiffer in their predicted effects of peripheral viewing onfairly straight ladders: Peripheral viewing should have asevere impact only in the class of models in which snakeand ladder associations are in competition with each other.In this study, we compared peripheral performance onsnakes and ladders.An unresolved issue relating to peripheral contour

detection is the extent to which alternating-phase snakesare detectable in the periphery. In this type of contour,the phase of the elements differs by 180- betweenneighboring elements along the contour. Detection ofalternating-phase contours in the periphery can be goodwhen the paths form smooth, closed, recognizableshapes, such as circles or ellipses (Kuai & Yu, 2006).In contrast, using contours that randomly changed direc-tion from element to element, Hess and Dakin (1997,1999) found a complete failure to detect alternating-phasesnakes at an eccentricity of 20- visual angle. However,Nugent et al. (2003) attempted to replicate Hess andDakin’s experiment as closely as possible and found thatperformance on alternating-phase snakes at 20- wasaround 70–80% correct and did not differ substantiallyfrom performance on same-phase snakes. The reasons forthis discrepancy are not clear. One possible explanationis that there were individual differences between thesubjects in the two studies. The small number of subjectsin these studies makes it likely that the results couldhave been affected by individual differences (Hess &Dakin, 1997, 1999, had two subjects, and Nugent et al.,2003, had three subjects). Further support for this idea isthat one of Hess and Dakin’s (1999) subjects (RH) hassubsequently shown chance performance in anotherexperiment, when a different subject in the same experi-ment was substantially above chance: Ledgeway et al.(2005) found that, for straight contours with elementsoriented at 45- to the path, subject RH (labeled RFH inLedgeway et al.’s paper) was at chance, whereas subjectTL obtained a score of around 70% correct. Thisdiscrepancy between RH and TL in Ledgeway et al.’sexperiment closely mirrors the discrepancy between Hessand Dakin’s results and those of Nugent et al. In thisstudy, we have restricted ourselves to same-phase snakesand ladders.

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 2

Methods

Subjects

Experiment 1 used four male subjects, BCH, BT,DHB, and KAM, who all had normal or corrected-to-normal vision. One subject, KAM, was aware of thepurposes of the study. All subjects were experiencedpsychophysical observers, but prior experience withcontour integration experiments ranged from virtuallynone to considerable experience. All subjects had hadapproximately equal experience with snake and laddercontours, making them ideal for this study, in whichperformance on the two contour types was being com-pared. BCH and KAM had had the most experience withcontour integration experiments, and these two subjectsparticipated in Experiment 2.

Apparatus

The experiments were run on a PC fitted with agraphics card with a DVI digital output (GeForce6800DDR 256 MB 256 bit PCI-E MS-8984-12S). TheDVI output was connected to a Cambridge ResearchSystems Bits++ processor, which combined the red andthe green channels, to produce a 14-bit digital grayscaleimage. The Bits++ processor then converted this 14-bitdigital image into an analogue signal, which wasdisplayed on a Sony CPD-520GS Monitor with aresolution of 1024 ! 768 pixels, and frame rate 85 Hz.The background luminance remained at 50.6 cd/m2

throughout the experiment. Images were calibrated byusing a Cambridge Research Systems OptiCAL luminancemeter to find the function that mapped 14-bit digitalvalues onto luminance levels; the inverse of this functionwas used to convert luminance levels to 14-bit digitalvalues, which were then distributed across the red and thegreen channels appropriately. Experiments were con-trolled using MATLAB (The MathWorks, Inc.), whichwas also used to create the images. Images were displayedusing the Psychophysics Toolbox (Brainard, 1997; Pelli,1997). Subjects viewed the screen binocularly from adistance of 100 cm.

Procedure

The procedure was very similar in the two experiments.There were two independent variables: contour type(snake or ladder) and eccentricity of fixation (Experiment1: 0-, 4-, or 6- visual angle; Experiment 2: 0-, 4-, 6-, or 8-visual angle). Snakes and ladders were randomly inter-leaved within sessions. Fixation eccentricity was blockedinto separate sessions. Subjects performed four sessions at

each eccentricity, fixating left, right, up, or down inseparate sessions, except for the foveal condition (0-eccentricity), in which subjects fixated at the center of thescreen for all four sessions. The order of sessions wasrandomized differently for each subject, except that thefirst session in each experiment for each subject was afoveal condition, so the subjects could get a good look atthe two types of contour before performing any peripheralcondition.Within each session, there were 50 trials for each

contour type. Each trial consisted of a 1000 ms fixationperiod, followed by two 247 ms intervals separated intime by 1000 ms: One randomly selected intervalcontained a contour stimulus, and the other contained astimulus with only distractor elements. The stimuli werealways centered on the center of the screen. The fixationperiod and interstimulus interval contained a smallfixation dot at the fixation location. The fixation dotwas a dark Gaussian blob with standard deviation 0.04-visual angle and contrast 0.5. For the foveal condition,the fixation dot disappeared during presentation of thestimuli, so that it did not interfere with them; forthe other conditions, the fixation dot stayed on for theduration of the trial to help the subject to maintaincorrect fixation. On each trial, the subject indicated witha mouse which interval he thought contained thecontour. After each trial, the subject received auditoryfeedback to indicate whether the response was correct orincorrect.The main difference between Experiments 1 and 2 was

that, in Experiment 1, the contour elements were perfectlystraight and jitter-free, whereas, in Experiment 2, thecontours had a small amount of curvature (a 10- pathangle between adjacent contour segments), and the pathangle, the element separation, and the orientation of theelements relative to the path were randomly jittered tobring foveal performance away from ceiling. The algo-rithm for generating the stimuli is described in the nextsection. Examples of stimuli from both experiments areshown in Figure 1.

Stimuli

There were four separate aspects of stimulus construc-tion: generating the Gabor elements, positioning thecontour elements, positioning the distractor elements,and choosing the orientations of the elements. These aredescribed in detail in the next four subsections.

The Gabor elements

The Gabor elements were generated using Equation 1:

L"x; y# $ L0"1% cw#; "1#

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 3

where c is the carrier, and w is the envelope, as defined inEquations 2 and 3, respectively:

c $ Csin&2: f "x cos E% y sin E#'; "2#

w $ exp( x2 % y2" #

2A2

! "; "3#

where L is the luminance at position (x, y), measured fromthe center of the Gabor patch; L0 is the mean (back-ground) luminance (50.6 cd/m2); C is the Michelsoncontrast, which was set to 0.5; f is the spatial frequency ofthe Gabor carrier, which was 4.65 cycles/deg; E is theorientation of the element; and A is the standard deviationof the Gaussian window and was set to 0.108-, that is, halfthe carrier wavelength.

Positioning the contour elements

The construction of the contour stimuli is illustrated inFigure 2. The parameters (shown in Table 1) differedbetween Experiments 1 and 2: In particular, the contoursin Experiment 1 were straight and jitter-free. The centersof the Gabor elements were positioned as follows. First, apath was generated, which formed the invisible

“backbone” of the contour. This consisted of a set ofeight invisible lines, of length d, called path segments,which were joined end-to-end. The orientation of eachsegment differed from that of the adjoining one(s) by arandomly selected angle of (! + $!) or (j! + $!), where! is a fixed path angle and $! is a small random value (thepath angle jitter). An element was placed on each pathsegment, at a distance $d from the center of the segment,where $d is a small random value (the separation jitter).The element separation is the distance between the centersof adjacent path segments (assuming no path angle jitter).The segment length, d, was set to achieve an elementseparation, s, of 0.645- visual angle (three times thewavelength of the Gabor Carrier), using the equation, d =s/cos(!/2). Once the contour had been generated, it wasshifted so that its midpoint (the junction between thesegments containing the fourth and the fifth elements) layexactly at the center of the display. In Experiment 1 only,the contour was then rotated so that its global orientationwas vertical (for left or right fixations) or horizontal (forup or down fixations), plus a small random value betweenT10-. The purpose of this constraint was to give a tighter

Figure 1. (A and B) Examples of the snake (A) and the ladder(B) stimuli used in Experiment 1. (C and D) Examples of the snake(C) and the ladder (D) stimuli used in Experiment 2. To makethese examples of snakes and ladders as comparable aspossible, the elements in panels B and D have the same positionsas those in panels A and C, respectively. Readers who havedif!culty seeing the contours can view Supplementary Figure 1, inwhich the contour elements have a higher contrast than thedistractor elements.

Figure 2. A schematic representation of part of a snake contour,illustrating the different parameters. For illustration purposes,the snake shown here has a higher curvature than used in theexperiments, and some of the jitter values are exaggerated. Thethick solid lines represent the invisible segments (of length d) thatform the backbone of the contour. A Gabor element waspositioned at the midpoint of each segment and then shiftedalong the segment by an amount $di (the separation jitter forelement i). The angle between each segment, i, and the next wasequal to T! (the path angle), plus a random jitter value, $!i. Thesign of the path angle was randomly determined for each junctionbetween segments. s is the element separation, that is, thedistance between the centers of adjacent path segments, assum-ing no path angle jitter. For snakes, each element, i, was alignedwith its segment before a jitter value, $Ei, was added to itsorientation. For ladders, the element was orthogonal to itssegment before adding the jitter value. In Experiment 1, the pathangle was 0-, and there was no jitter in any of the parameters; inExperiment 2, the path angle was 10-; the jitter values are given inTable 1.

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 4

control on the distance of the contour elements from thefixation point. In Experiment 2, there was no constraint onthe orientation of the contour as a whole: Because thesecontours were curved, it is not so straightforward to definewhat is meant by the “orientation” of the contour.

Positioning the distractor elements

An invisible grid (12 rows ! 12 columns) wasgenerated, centered on the center of the display. Forstimuli without a contour, a distractor element was placed

in each grid square; for stimuli with a contour, a distractorelement was placed in any grid square not occupied by acontour element. The position of each distractor wasdetermined randomly such that the horizontal and thevertical positions were not more than a quarter of a gridsquare width from the center of the square. An additionalconstraint was that the minimum allowed distancebetween the centers of any two elements was 4A, whereA is the standard deviation of the Gabor envelope.Because the elements were quite closely packed, thealgorithm would often reach a state in which it wasimpossible to place the next element in such a way that itwas sufficiently far away from other elements that hadalready been placed. If this situation arose, the algorithmwould reject the stimulus and start a new one.To avoid a density cue, we calculated the width of the

grid squares so that the mean separation between adjacentdistractor elements was equal to the element separation, s.If each distractor had been located exactly on the center ofthe grid square, the mean separation between adjacentdistractors would have been (1 + !2)W/2 , 1.207W, whereW is the width of the grid square (Beaudot & Mullen,2003). However, the jitter in distractor position was large,so we took account of the effect that this would have onthe mean distance. It proved difficult to derive a closed-form expression for the mean separation, taking distractorjitter into account, so we generated a million 3 ! 3 gridsof distractor elements, with positions randomized asdescribed above, and found the mean distance of thecentral element from the eight surrounding elements. Itwas found to be 1.225W, so the grid square width was setto s/1.225. Because the diagonal distance across a gridsquare is 1.414W, there was a small possibility that twocontour elements could fall within the same grid square. Ifthis happened, the stimulus was rejected. Thus, in everystimulus, there were exactly 144 elementsVone in eachgrid square.

Element orientations

For snakes, the orientation, E, of each contour elementwas set to (7 + $E), where 7 was the orientation of thatelement’s path segment, and $E was a random value (theorientation jitterVsee Table 1). The orientation of ladderelements was (7 + 90- + $E). The orientation of eachdistractor element was completely random.

Modeling

The simulations consisted of three stages. Firstly, all thestimuli used in the experiments were processed with asimple association field algorithm that formed contours bylinking pairs of elements. Secondly, each contour in eachstimulus was rated according to three criteria: number of

Parameter Value

Stimulus duration 247 msInterstimulus intervalduration

1000 ms

Number of contourelements

8

Total number ofelements

144 (12 ! 12 grid)

Michelson contrast 0.5Carrier spatial frequency(cycles/deg)

4.65

Carrier wavelength,1 (degrees ofvisual angle)

0.215

A (degrees ofvisual angle)

0.108

1/A 2Separation,s (degrees ofvisual angle)

0.645

s/1 3s/A 6Path segment length,d (degrees ofvisual angle)

Experiment 1: 0.645Experiment 2: 0.648

Path angle, ! Experiment 1: 0-Experiment 2: 10-

Path angle jitter, $! Experiment 1: noneExperiment 2: uniform probabilitybetween T5-

Separation jitter, $d Experiment 1: noneExperiment 2: uniform probabilitybetween T0.05d

Orientation jitter, $E Experiment 1: noneExperiment 2: uniform probabilitybetween T20-

Global contourorientation

Experiment 1: random value betweenT10- from vertical for left and right!xations, and T10- from horizontalfor up and down !xations.

Experiment 2: completely random

Table 1. Stimulus parameters. Each jitter value was selectedindependently for each element or path segment in a stimulus.

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 5

elements, straightness, and proximity to the center of thestimulus.1 Thirdly, the experiments were simulated,separately for each subject, by taking the two stimuli oneach trial and selecting as the target the stimulus that gaverise to the highest-rated contour.

The association !eld algorithm

We implemented an association field algorithm inwhich snake and ladder associations competed directlyfor ownership of the elements. We do not claim that ouralgorithm is a realistic model of the cortical processesunderlying the association/integration field: As Pelli et al.(2004) made clear, “findings to date provide only hints asto the nature of this computation” (p. 1138). Instead, oursimulations should be seen as a concrete confirmation ofour earlier intuitive argument that, when weak ladderassociations compete with strong snake associations,performance will be good on both types of contour whenthe association field is small (i.e., in the fovea) but will bemuch more greatly impaired for ladders than snakes whenthe association field is large (i.e., in the periphery).The inputs to the algorithm were the positional

coordinates and the orientations of the Gabor elements,rather than the images themselves. This is because, inPelli et al.’s (2004) model, the initial processing of theretinal image is performed, not by the integration fields,but by the earlier “feature detection” stage, which we donot address here. It seems that the representations offeatures in the integration field are somewhat symbolic,on/off representations, rather than the quasi-linear repre-sentations found in early vision. One piece of evidence forthis is that introducing very large variability in thecontrasts of the stimulus elements has a minimal effecton contour detection (Hess, Dakin, & Field, 1998).Additional evidence is that the function relating maskcontrast to magnitude of crowding is similar to thepsychometric function for letter identification perfor-mance as a function of contrast (Pelli et al., 2004,Figure 11); this suggests that whatever the target-maskspacing, as long as the mask is high-contrast enough to beidentified, it produces the same crowding effect as a maskof maximum contrast.For each element, the association field algorithm

assessed the strength of the snake and the ladderassociations between that element and every otherelement. The strength of the association between anytwo elements was calculated by multiplying together threescores, M, P, and C. M is a scaling factor that causessnake and ladder associations to have different strengths.For snake associations, M was set to 1, and for ladderassociations, M was less than 1, the exact value being afree parameter fitted to the data of each individual subject.P is a measure of the spatial proximity of the elements. Cscores the configuration of the pair of elements andmeasures the extent to which the pair of elements formspart of a perfectly straight snake or ladder contour.

The spatial proximity score, P, varies as a cosinefunction of the separation between the elements, S. If theelements in a pair are separated by a distance, S, and thespatial radius of the association field is r, then P for thatpair of elements is given by

P $ cos":S="2r## S e r0 S 9 r

#"4#

This function is plotted in Figure 3.It is clear from Equation 4 and Figure 3 that the

proximity score for a pair of elements depends on the sizeof the association field that is used to associate that pair ofelements. To explain how the association field radius, r,was determined for each pair of elements, we need to lookat Pelli et al.’s (2004) model in more detail. Theyproposed that, at each point in the visual field, there is arange of available integration field sizes, and the observerwill attempt to choose an integration field size that isappropriate for the task at hand. In our model, there is afree parameter (fitted separately to each individual sub-ject’s data) called the “preferred association field radius”;this parameter specifies the association field radius that themodel will choose, if available (we would expect thisparameter to be task and stimulus dependent but, becauseall our stimuli had the same element separation, we wereable to fit this parameter across all conditions of bothexperiments). Pelli et al.’s critical proposal was that thesmallest available association field radius at each retinallocation is proportional to the eccentricity so that, in theperiphery, the preferred association field radius may notbe available; in this case, the model will choose thesmallest available field radius. The model thereforerequires a further parameter, k, which is the rate at whichthe smallest available field radius increases with eccen-tricity: k is multiplied by the eccentricity to give the

Figure 3. The function relating the proximity score, P, to theseparation between the elements, S. r is the spatial radius of theassociation !eld.

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 6

smallest available field radius. Rather than fitting k to ourdata, we gave it a value of 0.5 because the critical spacingfor crowding to occur has already been found to be abouthalf the eccentricity (Bouma, 1970; Pelli et al., 2004; Toet& Levi, 1992). At locations for which the preferred radiuswas greater than half the eccentricity, the model used thepreferred association field radius; at all other locations, theassociation field radius was set to half the eccentricity.This meant that, for a given pair of elements, theassociation fields centered on the two elements wouldusually have different sizes. When calculating the associ-ation strength between two elements, the model alwaysused the larger of the two association fields.The configuration score, C, was a function of the extent

to which the pair of elements corresponded to a perfectsnake or ladder configuration. This was evaluated asfollows. First, the algorithm found the orientation differ-ence between each of the two elements and the linejoining them (see Figure 4); these two angles, "1 and "2,were expressed as positive values between 0- and 90-, andtheir average, ", was found. The ideal value of " (labeled"0) is 0- for snake associations and 90- for ladderassociations. The configuration score, C, varied as acosine function of the deviation of " from the ideal value:

C $cos 8j"j "0j

$ %% 1

& '=2 j"j "0j e 22:5-

0 j"j "0j 9 22:5-

(

"5#

This function is plotted in Figure 5.For each possible pair of elements, the algorithm found

the strength of the snake and the ladder associationsbetween them, by multiplying together the scores M, P,and C. All the associations were then sorted intodecreasing order of association strength (regardless ofwhether they were snake or ladder associations). Startingwith the strongest association, the algorithm worked its

way down the list; for each association, a link of theappropriate type (snake or ladder) was inserted betweenthe associated elements as long as none of the followingrules were broken:

1. an element can be linked to a maximum of two otherelements;

2. if an element is linked to two others, the change ofdirection at the junction between the two links shouldbe no more than 70- (this follows from Field et al.’s,1993, finding that performance is always at or veryclose to chance for path angles beyond this value);

3. if an element is linked to two others, both links mustbe of the same type (snake or ladder).

The algorithm continued to insert links according to theserules until the association strength was less than 1% of themaximum for that stimulus. Because the M score washigher for snakes, the snake-type links tended to beinserted first, preventing the linked elements from formingladder-type links.After the links between elements had been inserted,

contours were formed by taking elements with only onelink, and tracing along the links from element to elementuntil reaching another element with just one link.

Rating the contours

Each contour was given three scores (between 0 andapproximately 1) reflecting its length, straightness, andproximity to the center of the stimulus. These scores werethen multiplied together to give a single rating. Thus, acontour would have to have a high value on all threescores to get a good rating.The score for length was given by n/8, where n is the

number of elements in the contour. Thus, the shortestpossible contour (two elements) had a score of 0.25, and

Figure 4. (A) A perfect snake con!guration: "1 and "2 (not shown)are both 0-. (B) A con!guration that is neither a perfect snake nora perfect ladder. (C) A perfect ladder con!guration: "1 and "2 areboth 90-.

Figure 5. The con!guration score (C) for snake associations (redline) and ladder associations (blue line) as a function of ".

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 7

one with eight elements had a score of 1. Because thetarget contours all had eight elements, a perfectlyintegrated target would achieve a score of 1, but the scorecould be higher if a distractor element was linked to thecontour.The score for straightness was given by (70 j +)/70,

where + was the mean orientation difference (in degrees)between adjacent links along the contour. Seventy degreeswas the maximum possible value of + because theassociation field stage would not allow an element tohave links that differed in orientation by more than 70-.The score for proximity to the center of the stimulus

was given by (D j R)/D. R is the distance between thecenter of the stimulus and the element in the contour thatwas closest to the center. D is half the diagonal distanceacross the invisible 12 ! 12 grid into which the Gaborelements were inserted; this value (4.47- visual angle) wasthe maximum possible value of R.

Simulating the experiments

For each subject, the model was fitted to that subject’sfull set of percent-correct scores from Experiments 1 and 2(BCH and KAM) or Experiment 1 only (BT and DHB).The experiments were simulated separately for eachsubject, using the stimuli that that subject had seenduring the experiments (for modeling purposes, ratherthan using the actual stimulus images, we used the sets ofelement positions and orientations from which thoseimages had been generated). For each trial, we ran themodel on the two stimuli that the subject had seen on thattrial and selected as the target the stimulus with thehighest-rated contour. The model’s responses were clas-sified as correct or incorrect to obtain a percent-correct

score for each condition. The two free parameters (thepreferred association field radius and the M score forladders, which sets the strength of ladder associationsrelative to snake associations) were varied to find the bestfit to each subject’s data set.

Results

Psychophysical data

In both experiments, the data from the four sessions (up,down, left, and right fixation) on each eccentricity andcontour type were pooled to give 200 trials per condition.The results of Experiments 1 and 2 are shown in Tables 2and 3, respectively, and Figures 6 and 7.Two kinds of statistical analysis were performed on the

data. One analysis was a one-tailed binomial test of thenull hypothesis that the subjects were equally likely to becorrect or incorrect on each trial. A one-tailed test wasused because we would not expect performance to bebelow chance. This test was carried out for each individualcondition, and the p values are shown in Tables 2 and 3.There were multiple tests for each subject, and thecriterion of significance for each test, !t, was chosen suchthat the type I error probability for the whole experiment,!e, was equal to .05. If there are n tests, then

!t $ 1j "1j !e#1=n: "6#

There were six tests per subject in Experiment 1 and eighttests per subject in Experiment 2, which gave a criterion

Subject Eccentricity Snake Ladder Chi-square test

BCH 0 99.5 (p = 1.3 ! 10j58)* 99.5 (p = 1.3 ! 10j58)* #2 = 0 (p = 1)4 99.5 (p = 1.3 ! 10j58)* 81 (p = 9.9 ! 10j20)* #2 = 38.9 (p = 4.5 ! 10j10)*6 99.5 (p = 1.3 ! 10j58)* 52 (p = .31) #2 = 123 (p G 2 ! 10j16)*

BT 0 99 (p = 1.3 ! 10j56)* 98 (p = 4.1 ! 10j53)* #2 = 0.677 (p = .41)4 98 (p = 4.1 ! 10j53)* 59 (p = .0066)* #2 = 90.1 (p G 2 ! 10j16)*6 97.5 (p = 1.6 ! 10j51)* 45.5 (p = .11) #2 = 133 (p G 2 ! 10j16)*

DHB 0 100 (p = 6.2 ! 10j61)* 100 (p = 6.2 ! 10j61)* -4 100 (p = 6.2 ! 10j61)* 57.5 (p = .02) #2 = 108 (p G 2 ! 10j16)*6 99.5 (p = 1.3 ! 10j58)* 52.5 (p = .26) #2 = 121 (p G 2 ! 10j16)*

KAM 0 99.5 (p = 1.3 ! 10j58)* 100 (p = 6.2 ! 10j61)* #2 = 1.00 (p = .32)4 100 (p = 6.2 ! 10j61)* 50.5 (p = .47) #2 = 132 (p G 2 ! 10j16)*6 100 (p = 6.2 ! 10j61)* 53 (p = .22) #2 = 123 (p G 2 ! 10j16)*

Table 2. Results of Experiment 1. The snake and ladder columns give the percent correct for each condition. The p values in thesecolumns are the results of one-tailed binomial tests of the null hypothesis that the subject was equally likely to be correct or incorrect oneach trial. Asterisks indicate signi!cant p values. The criterion of signi!cant for each binomial test was .0085, giving a per-experimentcriterion of .05. The chi-square test column gives the results of the chi-square tests comparing performance on snakes and ladders ateach eccentricity. The criterion of signi!cance on each of these tests was .0170. Again, asterisks indicate signi!cant p values.

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 8

of significance of .0085 for each individual test inExperiment 1 and .0064 in Experiment 2.In the other analysis, we compared performance on

snakes and ladders at each eccentricity, using a methodsuggested by Lovell (2002). For each eccentricity, eachtrial was classified according to two variables: snake/ladder and correct/incorrect. This yielded a 2 ! 2contingency table, on which a chi-square test could beperformed to see if the contour type had had a significanteffect on the proportion of correct responses. Separate chi-square tests were carried out at each eccentricity for eachsubject. There were three chi-square tests per subject inExperiment 1, and four in Experiment 2; using Equation 6,this give criteria of significance per test of .0170 and .0127in Experiments 1 and 2, respectively.

Experiment 1

In Experiment 1 (Table 2, Figure 6), the contours werecompletely straight. Performance on snakes was at or closeto ceiling at all eccentricities. Performance on ladders wasat ceiling in the foveal condition but was not significantlydifferent from chance at the 6- eccentricity for any of thesubjects. At the 4- eccentricity, only two subjects showed

ladder detection performance that was significantly betterthan chance. The chi-square tests showed that, on allnonfoveal conditions, performance on snakes was signifi-cantly better than ladders for all subjects.Although the effects of peripheral viewing were very

different for snakes and ladders, Experiment 1 does notprovide unequivocal support for the view that detection ofladders is disproportionately impaired in the peripherybecause performance on both contour types was at ceilingin the fovea. There might instead have been a generalimpairment for ladders, which only showed up in theperiphery, where the performance on ladders fell awayfrom ceiling. To rule out this possibility, subjects BCHand KAM also performed Experiment 2, in which thecontours were jittered to bring foveal performance awayfrom ceiling.

Experiment 2

For both subjects in Experiment 2, the performancelevels on snakes and ladders were very closely matchedaway from ceiling in the foveal condition but diverged inthe eccentric conditions (see Table 3 and Figure 7). Theseresults suggest that detection of ladder contours is indeed

Figure 6. Results of Experiment 1. The red and blue symbols show the psychophysical data (given in Table 2) from snake and ladderconditions, respectively; !lled symbols represent scores that are signi!cantly different from chance, and open symbols represent chance-level performance. The red and the blue lines show the performance of the best-!tting models.

Subject Eccentricity Snake Ladder Chi-square test

BCH 0 82.5 (p = 1.1 ! 10j21)* 81.5 (p = 2.3 ! 10j20)* #2 = 0.0678 (p = .79)4 70 (p = 7.5 ! 10j9)* 56 (p = .052) #2 = 8.41 (p = .0037)*6 65.5 (p = 6.9 ! 10j6)* 55.5 (p = .069) #2 = 4.18 (p = .041)8 63 (p = 1.4 ! 10j4)* 53.5 (p = .18) #2 = 3.71 (p = .054)

KAM 0 92 (p = 1.2 ! 10j37)* 93 (p = 7.9 ! 10j40)* #2 = 0.144 (p = .70)4 71.5 (p = 5.1 ! 10j10)* 55 (p = .089) #2 = 11.7 (p = 6.2 ! 10j4)*6 73 (p = 2.8 ! 10j11)* 51 (p = .42) #2 = 20.5 (p = 5.8 ! 10j6)*8 75 (p = 4.2 ! 10j13)* 51.5 (p = .36) #2 = 23.8 (p = 1.1 ! 10j6)*

Table 3. Results of Experiment 2, presented in the same way as for Experiment 1 (Table 2). The criterion of signi!cance was .0064 foreach binomial test and .0127 for each chi-square test.

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 9

disproportionately impaired in the periphery, comparedwith snakes. Performance on ladders dropped to chancelevel at 4- eccentricity, whereas performance on snakesremained significantly above chance at all eccentricities.The chi-square tests showed that, for KAM, snakedetection was significantly better than ladder detection atall eccentricities except the foveal condition. For BCH,only the 4- eccentricity showed a significant difference:Although snake detection performance at the 6- and 8-eccentricities was significantly above chance (unlikeladder detection), it was not quite high enough to bestatistically different from the ladder detection perfor-mance on these conditions.

Modeling

The best-fitting parameter values for each subject aregiven in Table 4. The model’s performance is shown inFigures 6 and 7; the numerical values are given in theauxiliary file, SimulationResults.txt. The model fitted verywell to the data from Experiment 1. Performance onsnakes was 100% correct at each eccentricity; perfor-mance on ladders was close to 100% correct in the foveabut dropped quickly to chance in the periphery. The modelfitted less well to the data from Experiment 2 but stillshowed the right kind of pattern: Performance was quiteclosely matched for snakes and ladders in the fovea(although not as closely matched as in the psychophysicaldata); in the periphery, performance on ladders dropped tochance at 4- eccentricity, whereas performance on snakesdropped more gradually. For snakes, the model’s perform-ance fell with increasing eccentricity at a very similar rateto BCH’s data, although BCH consistently performedworse than the model.To give an insight into the workings of the model,

Figure 8 shows the results of processing the examplestimuli shown in Figure 1. In the fovea, where theassociation fields are small, the target contours all get higher

ratings than the spurious background contours. In theperiphery, where the association fields are larger, the ladderassociations cannot compete with spurious snake associa-tions, and the target ladder contours are not detected.

A demonstration of crowding

The central argument in this paper is that the same kindof mechanism underlies both contour integration andcrowding. To demonstrate that our contour integrationmodel shows crowding-like behavior, we applied ourmodel to letter stimuli, composed from short line seg-ments (shown in Figure 9). The results are shown inFigure 10. We assumed that crowding had occurred ifcontours had formed between the letters, joining themtogether. The model correctly predicts three key charac-teristics of crowding, identified by Pelli et al. (2004): Thecritical spacing for crowding to occur is independent ofthe size of the target letter, scales with eccentricity, and isgreater on the peripheral side of the target.The critical spacing has been found to be about half the

eccentricity (Bouma, 1970; Pelli et al., 2004; Toet & Levi,1992). Our model does show some crowding at largerspacings than this, but in these cases, interference is onlycaused by the more peripheral flanker. A single flanker ismuch less effective at producing crowding than a pair offlankers (Pelli et al., 2004), presumably because the letteridentification mechanisms are reasonably tolerant tointerference from just one side. Thus, we might predictthat the crowding effect from the peripheral flanker alonewould be relatively small: Crowding would be more likelyto be detected when both flankers interfered with thetarget and, in our model, this occurs when the spacing ishalf the eccentricity, as found psychophysically.

Discussion

Experiment 1 showed that straight ladders are easy todetect in the fovea but are undetectable at eccentricities of

Subject

M scorefor

ladders

Preferredassociation

field radius (-)

RMS error in% correctscores

BCH 0.61 0.87 7.90BT 0.475 0.775 1.79DHB 0.44 0.74 1.66KAM 0.57 0.83 6.79

Table 4. Best-!tting parameters and root mean square (RMS) erroracross Experiments 1 and 2 (BCH and KAM) or Experiment 1 only(BT and DHB). The M score for ladders sets the strength of ladderassociations relative to snake associations.

Figure 7. Results of Experiment 2, plotted in the same way as forExperiment 1 (Figure 6).

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 10

around 4- to 6- visual angle. In contrast, performance onstraight snakes was at or close to 100% correct at all theeccentricities that we tested. Experiment 2 showed that

this difference between snakes and ladders did not resultsimply from a general deficit in detection of ladders:When snakes and ladders were matched below ceiling inthe fovea, ladder detection fell to chance at an eccentricityof 4-, whereas snake detection remained significantlyabove chance up to the highest eccentricity that we tested(8-). The most striking comparison is perhaps betweenstraight ladders (Experiment 1) and jittered snakes(Experiment 2). Performance on jittered snakes was abovechance at each eccentricity; straight ladders were mucheasier to detect than jittered snakes in the fovea but wereundetectable in the periphery.Our results may partly explain the relative difficulty in

detecting ladders that has been reported in previousstudies (Bex et al., 2001; Field et al., 1993; Hess et al.,2000; Ledgeway et al., 2005): In all of these studies, theposition of the contour was randomized to some extent.The difference in the effect of eccentric viewing on snakesand ladders means that the positional randomizationwould have caused a greater disruption to detection ofladders. Experiment 2 shows that performance on snakesand ladders can be very similar if the contours are fairlystraight and positioned exactly at the fixation point.We were surprised to find such a severe deficit of ladder

detection at such small eccentricities. Ladders would beexpected to give rise to extended regions of response in a“second-order” or “non-Fourier” channel, of the kind

Figure 9. The top row (in black) shows the letter stimuli used inour crowing demonstration; the bottom row (in red) shows thecontours that the model detected in these letters when they werepresented in the fovea. Only snake contours were detected.

Figure 8. The results of processing stimuli A, B, C, and D of Figure 1 with our association !eld algorithm, with parameters that !tted best tothe data from subject KAM. Snake contours are shown in red, and ladder contours are shown in blue. The width of each contour’s line isproportional to that contour’s rating. The left half of this !gure shows the results when the image is centered on the fovea: The snake andthe ladder target contours are all detected. The right half of the !gure shows the results when the image is centered 6- to the right of!xation. Because of the larger association !elds, many more spurious contours are formed. The target snake contours still have higherratings than the spurious contours, whereas the target ladder contours are not detected.

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 11

proposed in many models of texture segregation (Graham,1991; Graham, Beck, & Sutter, 1992; Graham & Sutter,1998; Graham, Sutter, & Venkatesan, 1993; Lin &Wilson, 1996; Sutter, Beck, & Graham, 1989; Wilson,1993). In these models, a small-scale linear filter isfollowed by a nonlinearity (usually squaring or full-waverectification), followed by a large-scale linear filter withorientation orthogonal to that of the first filter. Such amechanism gives a strong response at a texture border(such as a border between two areas of differentorientation) but a weak response elsewhere. This kind ofmechanism could be used to detect fairly straight ladders:A small-scale filter aligned with the elements will givestrong positive and negative responses along the path ofthe contour; if these responses are rectified, then a large-scale filter orthogonal to the small-scale filter will give a

positive response along the whole of the contour. Theperiphery is known to contain many neurons that respondto second-order signals (Zhou & Baker, 1994, 1996), andsecond-order signals are certainly detectable in theperiphery (Hess, Baker, May, & Wang, 2007), so wemight have expected subjects to be able to detectperipheral ladder contours.Our preferred explanation is that the failure to detect

ladders in the periphery is a form of crowding. Pelli et al.(2004) proposed that crowding is caused by inappropriatefeature integration by large integration fields in theperiphery. We have shown that the same kind ofmechanism can account for our results. We simulatedour experiments with an association field model in whichthe minimum association field size at each retinal locationwas half the eccentricity, as suggested by the crowding

Figure 10. Contours formed when we applied our association !eld algorithm to letter stimuli, composed of short line segments. The lettershad a height of 1- visual angle (panels A and C) or 0.5- (panels B and D). They were separated by 2-, 4-, or 8-. Each row of letters E, F, andA was presented separately to the model. The center of the letter “F” was positioned 4- to the right of !xation (panels A and B) or 8- to theright (panels C and D). The model was the same as in Figure 8, except that the preferred association !eld size was set to 0.1- to be moreappropriate for perception of letters of these sizes. The critical spacing for crowding to occur is independent of the size of the target letter,scales with eccentricity, and is greater on the peripheral side of the target. Panels E–H are the same as panels A–D, except that theidentities of the two "anking letters have been swapped. These simulations con!rm that it is the positions of the "ankers, not theiridentities, that gave rise to the patterns of crowding shown in panels A–D.

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 12

literature. In our model, the only difference between theprocessing of snakes and ladders was that ladder associ-ations were about half as strong as snake associations (seeTable 4). It may be that future work will reveal furtherdifferences between the processing of snakes and ladders,but our simulations show that, if the association field sizescales with eccentricity, a difference in strength of snakeand ladder associations is sufficient to explain the verydifferent effects of peripheral viewing on these two typesof contour.Our current lack of knowledge about the mechanisms of

crowding makes it difficult to provide direct evidence thatcrowding causes the poor performance on ladders in theperiphery. However, we feel that the indirect evidence thatwe have presented here is reasonably strong. Our modelexplained the results of our contour integration experi-ments and also showed three key characteristics ofcrowding with letter stimuli: Critical spacing was inde-pendent of the size of the target letter, scaled witheccentricity, and was greater on the peripheral side ofthe target. Our proposal also fits with the subjectiveexperience of viewing contour stimuli in the periphery: Aswith crowding, it is easy to detect the presence of theelements within a peripheral ladder stimulus but difficultto interpret them.Our model fits very well to the data from Experiment 1

but deviates a little from those of Experiment 2. The mainproblem is that the model tended to overestimate theperformance level on snakes in Experiment 2. This mayhave arisen for two reasons: (1) In the experiment, thestimuli were presented for a short time (247 ms), tominimize eye-movements, and this may have given thesubjects insufficient time to process the stimuli fully; incontrast, the computer model had as much time as itneeded. (2) No noise was implemented in the model; theorientation and position of each Gabor element wereassumed to be represented with perfect accuracy, whereasa more realistic model would add noise to theserepresentations, reducing the model’s performance.In the Introduction, we outlined two possible types of

association field model. In one type, snake and laddercontours are integrated using completely independentmechanisms; in the other, snake and ladder associationscompete directly for ownership of the elements. In ourmodel, it is the competition between snakes and laddersthat leads to the very poor performance on ladders inthe periphery. If we removed this competitive aspect sothat snakes and ladders were processed by independentmechanisms, then the model would detect snakes andladders equally well. This is because the model relieson relative strength of associations rather than theabsolute values: Scaling all the ladder associationstrengths by the same factor would have no effect ifladders were processed by an independent mechanism.Our data therefore favor association field models inwhich snake and ladder associations are in directcompetition.

Our model of peripheral contour integration is some-what different from the model proposed by Hess andDakin (1997, 1999). Hess and Dakin proposed that there isan absence of explicit contour linking in the periphery,and that contour integration occurs due to the spatialoverlap of filter responses within individual orientationchannels. This model was rejected by Lovell (2002,2005): He showed that the performance of Hess andDakin’s model increases with the number of points ofinflection in a contour, whereas human observers gener-ally show the opposite effect in both fovea and periphery.Our current proposal is that explicit contour linking ispresent in the periphery but occurs over a wider area,allowing more distractors to interfere with the process. Inthese circumstances, contour integration would be con-sistently successful only in the most favorable conditions,that is, when the contour was fairly straight and theelements were collinear.Our current model would not show an effect of the

number of points of inflection. This behavior could easilybe added to the model, but we did not do so here becausethis would have added extra parameters which would nothave been constrained by our data: The contours inExperiment 1 were perfectly straight, so they had nopoints of inflection, and those in Experiment 2 had a pathangle that was too small to give rise to a large effect of thenumber of points of inflection.In this paper, we have not addressed the issue of the

effect of phase alternation on the detection of peripheralcontours. Given the very different results obtained by Hessand Dakin (1997, 1999) and Nugent et al. (2003), we feelthat this issue can only be resolved by a large studyfocusing on this issue. Our model is agnostic about theeffects of phase because phase is not represented in themodel: Only the position and orientation of each Gaborelement are represented. It would probably be possible toaccommodate the apparent individual differences in theeffect of phase alternation by introducing a parameter thatcontrolled the relative strengths of associations betweensame-phase and opposite-phase elements.Our results shed some light on the computations

performed by the integration/association fields. Firstly,our results suggest that the associations between collinearelements are stronger than those between parallel ele-ments. Secondly, these two types of association appear tobe in direct competition with each other, suggesting thatthey are processed by the same mechanism, rather thanindependent mechanisms (the view that snakes andladders are processed by the same mechanism is alsosupported by May & Hess’s, 2007, finding that snakes andladders are integrated at similar speeds). Thirdly, ourresults support Pelli et al.’s (2004) proposal that, at eachpoint in the visual field, the minimum integration/association field size is proportional to eccentricity.Because of the large size of contour stimuli, future modelsof contour integration should take into account the scalingof association field size with eccentricity, even when

Journal of Vision (2007) 7(13):9, 1–15 May & Hess 13

modeling the detection of foveally presented contours. Itis hoped that future work delineating the successes andfailures of feature integration in the periphery will revealmore details about the integration/association fields,which appear to play a crucial role in any visual taskmore complex than detection of basic features.

Conclusions

Straight ladder contours were easily detectable in thefovea but were undetectable at eccentricities of around 4-to 6-. Performance on straight snakes was at ceiling at alleccentricities that we tested. An association field modelcan account for these results if it has three key properties:(1) Snake associations are stronger than ladder associa-tions; (2) snake and ladder associations compete againsteach other for ownership of the elements, rather thanbeing processed by independent mechanisms; (3) theminimum association field size is proportional to theeccentricity. The last of these properties has beenproposed to account for the phenomenon of crowding:Large integration fields in the periphery cause inappro-priate integration of features from different parts of thestimulus (Pelli et al., 2004). We have shown that the samekind of process could underlie the failure to detect laddersin the periphery. We conclude that the failure to detectperipheral ladders is a form of crowding.

Acknowledgments

This work was supported by NSERC Grant no. RGPIN46528-06.

Commercial relationships: none.Corresponding author: Keith A. May.Email: keith@keithmay.org.Address: Vision Sciences Research Group, School of LifeSciences, University of Bradford, Richmond Road, Brad-ford, West Yorkshire, BD7 1DP, UK.

Footnote

1

The first two criteria were based on the empiricalfinding that that performance improves with increasingcontour length (Dakin & Hess, 1998) and straightness(Field et al. (1993); the third criterion was includedbecause the subjects were told that the contours passedthrough the center, so they could use this knowledge toperform the task.

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